We assume
$f(x,U)$ to be a convex set with $C^1$ boundary for all
$x\in\mathbb{R}^n$ and the target $\kappa$ to satisfy an interior
sphere condition.
For such problems we prove necessary and sufficient
optimality conditions
using the properties of the minimum time function $T(x)$.
Moreover, we give a local description of the singular set of
$T$.